This application is for a renewal of our previous grant entitled "Stochastic Models in Medicine and Biology." Our major interest remains with the analysis of a variety of population genetic models and related data analysis especially the interpretation of observed data in human and other populations. The emphasis is both on a more penetrating analysis of some of the important classical models and on making progress on the more complex models (employing a combination of mathematical and computing methodology) with the objective to interpret observed data and gain insight into the genetic structure of populations and evolutionary mechanisms. We plan to concentrate our efforts in four main areas: (1) The study of multigenic systems incorporating a variety of linkage relationships and selection regimes. (2) The nature of population structure under selection-migration interactions. (3) Aspects of the theory of polygenic and cultural transmission. (4) A new phase of our research will be intensive collaborative studies in methods of data analysis in two applications of medical genetics. The two applications chosen are (1) the HLA system and several associated diseases and (2) the data of the lipid research clinic at Stanford. It is our intention to bring to bear our knowledge of complex genetic models and statistical methodology for the evaluation of familial risk analysis. The interplay between theoretical analyses, data analyses, simulation studies and an awareness of the biological problems has been a key factor in our program. The unique collaborative efforts between the Biology Department (Professor M. Feldman) and the Genetics Department (Professor L. Cavalli-Sforza) and the Mathematics Department (Professors S. Karlin and J. McGregor) has provided an ideal setup in achieving several of the research objectives of the Grant. Professor W. Bodmer of Oxford, who is closely associated with our group, will continue as a regular consultant. A more detailed description of our anticipated research program follows.